#3333319

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(9,10)(11,12)(13,21)(14,15)(16,18)(17,24)(19,23)(20,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(6,8)(10,15)(11,18)(17,21)(19,22)(25,26,29,32)(27,33,35,30)(28,31,38,36)(34,40,37,39);(5,6)(7,8)(9,11)(10,16)(12,15)(13,22)(14,18)(17,20)(19,24)(21,23)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,4)(9,12,14,16)(10,11,15,18)(13,23,24,20)(17,22,21,19)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(9,13)(10,17)(11,19)(12,20)(14,24)(15,21)(16,23)(18,22)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)

Polycyclic presentation:

2^9|b^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3240515);